Facebook Inc.'s board of directors threw its support behind Chief Operating Officer Sheryl Sandberg on Wednesday, saying that it was "entirely appropriate" for her to ask if George Soros had shorted the company's stock after the billionaire investor called the social-media giant a "menace."

Facebook's general counsel Colin Stretch sent a letter from the board to Patrick Gaspard, president of Mr. Soros's Open Society Foundations. Mr. Gaspard has been among those who have criticized or questioned Ms. Sandberg's handling of the matter.

"To be clear, Ms. Sandberg's question was entirely appropriate given her role as COO," Mr. Stretch wrote. "When a well-known and outspoken investor attacks your company publicly, it is fair and appropriate to do this level of diligence."

The letter was a show of support for Ms. Sandberg, one of the tech industry's most prominent executives, as investors, regulators, and users renew pressure on both her and the company following a series of management missteps.

Facebook CEO Mark Zuckerberg is chairman of the board of directors.

Mr. Stretch sent the message at the Facebook board's request in response to a letter Mr. Gaspard publicized Tuesday. In that communication, Mr. Gaspard said that Ms. Sandberg had misled him during a private phone call on Nov. 15 about her knowledge of company-commissioned opposition research into Mr. Soros by Definers Public Affairs, a consulting firm that Facebook tasked with scrutinizing detractors.

During that call, Ms. Sandberg "assured me that she knew absolutely nothing about these efforts, which included oppo research on George Soros," Mr. Gaspard wrote in a his Dec. 4 letter to the company's board of directors. "It is now clear from the news that Ms. Sandberg did not speak in good faith."

On Wednesday, Mr. Stretch disputed Mr. Gaspard's characterization, saying Ms. Sandberg had been truthful to him about her knowledge of Definers's work regarding Mr. Soros. Open Society didn't immediately return a request for comment.

The exchange between Mr. Soros's group and Facebook is the latest twist in a difficult two years for the social-media company, where employee morale has collapsed as the company has faced one controversy after another.

Mr. Gaspard's comments weren't even the only firestorm this week. Earlier Wednesday, British lawmakers shared 250 pages of internal Facebook documents showing how company executives, including Mr. Zuckerberg, sought to leverage its developer platform.

Also Wednesday, Stifel analysts downgraded Facebook shares to a hold, citing the company's internal and political issues. "Facebook's management team has created too many adversaries -- politicians/regulators, tech leaders, consumers, and employees -- to not experience long-term negative ramifications on its business," the analysts wrote.

Ms. Sandberg's involvement in the Soros matter has been under scrutiny since The New York Times reported last month that Facebook had hired Definers in part to push back against the company's critics.

On Nov. 15, Ms. Sandberg said she didn't know about Definers and the work the firm was doing for the company.

The following week, she revised her statement to say that some of Definers's work was "incorporated into materials presented to me and I received a small number of emails where Definers was referenced."

Last week, Facebook confirmed that Ms. Sandberg sent an email to her staff in January asking whether Mr. Soros has shorted the company's stock after the liberal philanthropist called internet "monopolies" a "menace" in a speech at the World Economic Forum.

Facebook officials had already asked Definers to look into Mr. Soros when Ms. Sandberg sent her email, according to a Facebook spokesman.

"She told you, truthfully, that she had not known about the work done by the Definers public relations firm related to Mr. Soros," Mr. Stretch wrote Wednesday, adding that Ms. Sandberg takes "full responsibility" for work done under her watch.

The letter also outlines various steps Facebook has taken since the 2016 U.S. presidential election to secure its platform from bad actors trying to spread disinformation, including by setting up an independent election commission to explore the relationship between social networks and democracy.

"Much additional work remains, but to be clear these are not the actions of a company that has, in your words, 'failed to take responsibility'," Mr. Stretch wrote.